tomclarke wrote: Teahive wrote: So in earth FOR the muon clock ticks more slowly and it reaches earth before decaying, whereas in muon FOR clocks on earth would appear to tick slowly and the muon decays before it reaches earth?
No in muon FOR the distance from generation to earth is much smaller than in earth FOR. (Sorry, I should have said this earlier).
In fact it is the
actual distance and not the relativistic -distorted distance as observed from the earth. Look at it in this way. Within the muon's FOR (let me also use this term) the decay time of the muon is its actual decay time (tau) as measured within any laboratory. Furthermore, since the muon is stationary, the earth is approaching it at a speed v: Thus if the muon reaches the earth and then decays, the actual height above earth h at which it has been created must be h=v*(tau).
But when looking from earth, the space and time coordinates are distorted; and to obtain this distortion one must use the Lorentz transformation. When synchronizing the clock on earth with the muon’s clock when the muon reaches earth and decays, one finds that the corresponding space and time coordinates at which the muon has been created relative to earth are -H so that H>h and -(TAU) so that (TAU)>(tau): One thus has that H=v*(TAU). Thus the relativistic effect is that
both the decay time and the height are observed to lengthen; as they must do since the speed with which the earth and the muon approach each other is the same. There is no length contraction involved.
tomclarke wrote:To compare the times of two clocks they need to be co-located at two different times. If there is relative velocity you then need change in FOR (from acceleration) for the colocation ever to be repeated. This will either be symmetric - in which case no difference - or asymmetric, in which case the FOR change accounts for the time difference.
Teahive wrote:But if the clocks are not just co-located but stationary relative to each other at both comparison events, doesn't the situation have to be symmetric?
tomclarke wrote:No, because the travelling twin would experience acceleration, change of FOR turnaround acceleration, change of FOR, deceleration, change of FOR, wheread the "stationary twin would have no acceleration.
Consider the two twins within two space ships that start to move at the instant in time when the clocks of the two twins are synchronized. Each one accelerates at the same rate and then decelerates at the same rate: But since the speed that is generated is a relative speed, the acceleration must also be a relative parameter. The first spaceship accelerates at the same acceleration relative to the other spaceship as the other spaceship accelerates to the first. Thus one can choose either one of the spaceships to be stationary and the other one as accelerating. If after a while each decelerates they will eventually reach a relative speed of v=0. They then accelerate and decelerate to reach other. When they reach each other the relative speed will again be zero. Each twin will have observed the same total change in time on the other’s clock. Thus their clocks must show the same time when they meet up again.
Now let only one space ship accelerate (go in free fall) relative to the other space ship, then decelerate to come to a halt and accelerate back and decelerate to come to a halt at the space ship that supposedly did not accelerate. But as far as each clock is concerned it has been the other clock that accelerated away from it and returned. Thus the clocks must again show the same time. This is also borne out by Einstein’s postulate that an accelerating reference frame is also an inertial reference frame. So now we have a contradiction: This is so since Einstein also postulated that acceleration and gravity is the same.
So which is it? Is an accelerating reference frame an inertial reference frame as it must be for sanity to prevail when it comes to the ages of the two twins; or is acceleration and gravity equivalent? If the latter is the case, then an accelerating reference frame cannot be an inertial reference frame in the sense that it is equivalent to a non-accelerating inertial reference frame.
It is my opinion that the slowing down of time within a gravitational field is not a result of acceleration, but that it relates to the wave nature of matter. This would mean that Schroedinger essentially unified gravity with “quantum” mechanics when he postulated his differential wave equation.